Physics in a nutshell

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Unit Cell, Primitive Cell and Wigner-Seitz Cell

In the previous article we have seen how a crystal can be defined formally. Now some more concepts will be introduced which will be helpful for the visualisation and classification of crystal structures.

As mentioned before, the ideal crystal is an infinite object which thus cannot be illustrated as a whole in a single image. Even if one could, this would not be desirable. Instead one can use the fact that a crystal is built up by smallest elements that are repeated in all directions, filling the whole space.

Unit Cell

These elements are called unit cells and fulfil the following requirements:[1] [2]

  • A repetitive arrangement (pure translation) of them can build up the whole crystal without overlaps/gaps.
  • There is no further partition of the unit cell that could itself be used as a unit cell.

One finds varying definitions of this term in the literature.[3] [4] On this website the definition above will be used consistently.

For a given crystal there are always quite a few possible unit cells:

Different unit cells for a certain lattice.
For a certain lattice there are many different possible unit cells. Here are some examples for a two-dimensional lattice.

Often it is convenient to choose the one with the highest level of inner symmetry.

Primitive Cell

A primitive cell is a unit cell that contains exactly one lattice point. It is the smallest possible cell.[5] If there is a lattice point at the edge of a cell and thus shared with another cell, it is only counted half. Accordingly, a point located on the corner of a cube is shared by 8 cubes and would count with $\frac{1}{8}$.

Wigner-Seitz Cell

There is a special type of primitive-cells known as Wigner-Seitz cell. The Wigner-Seitz cell of a lattice point is defined as the volume that encloses all points in space which are closer to this particular lattice point than to any other. It can be constructed as depicted below.[6] [7]

Wigner-Seitz cell construction
2D-construction of a Wigner-Seitz cell: One chooses any lattice point and draws connecting lines to its closest neighbours. In a second step one constructs the perpendicular bisectors of the connecting lines. The enclosed area is the Wigner-Seitz cell. It forms a unit cell, i.e. is able to build the whole lattice without gaps/overlaps.[8]

References

[1] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 1.1.1.1)
[2] S. H. Simon Oxford Solid State Basics Oxford 2013 (p. 114)
[3] S. Hunklinger Festkörperphysik De Gruyter 2014 (p. 55)
[4] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 1.1.1.2)
[5] S. Hunklinger Festkörperphysik De Gruyter 2014 (p. 56)
[6] G. Iadonisi, G. Cantele, M. L. Chiofalo Introduction to Solid State Physics and Crystalline Nanostructures Springer 2014 (ch. 1.2.1)
[7] S. H. Simon Oxford Solid State Basics Oxford 2013 (p. 115)
[8] Ch. Kittel Einführung in die Festkörperphysik Oldenbourg 2006 (p. 7)

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